LANGUAGE AND GRAMMARS © University of LiverpoolCOMP 319slide 1

Contents Languages and Grammars Formal languages Formal grammars Generative grammars Analytic grammars Context-free grammars LL parsers LR parsers Rewrite systems L-systems © University of LiverpoolCOMP319slide 2

Software Engineering Foundation Software engineering may be summarised by saying that it concerns the construction of programs to solve problems and that there are three parts:  Construction/engineering, and methods  Problems, and problem solving, and  Programs © University of LiverpoolCOMP319slide 3

Languages and grammar Languages are spoken and written (linguistics) To be effective they must be based on a shared set of rules – a grammar Grammars are introspective they are based on and couched in language Natural language grammars are constantly shifting and locally negotiated A grammar is a formal language in which the rules of discourse are discussed and are the aim © University of LiverpoolCOMP319slide 4

Formal language concepts The concept emerges because of the need to define rules (for language) Formally, they are collections of words composed of smaller, atomic units Issues of concern are  the number and nature of the atomic units,  the precision level required,  the completeness of the formalism © University of LiverpoolCOMP319slide 5

Examples of formal languages The set of all words over {a, b} The set {a n : n is a prime number} The set of syntactically correct programs in a given computer programming language The set of inputs upon which a certain Turing machine halts © University of LiverpoolCOMP319slide 6

Formal language specification There are many ways in which a formal language can be specified e.g. strings produced in a formal grammar strings produced by regular expressions the strings accepted by automata logic and other formalisms © University of LiverpoolCOMP319slide 7

Language Production Operations Concatenation of strings drawn from the two languages Intersection or union of common strings in both languages Complement of one language Right quotient of one by the other Kleene star operation on one language Reverse of a language Shuffle combination of languages © University of LiverpoolCOMP319slide 8

Formal Grammars Noam Chomsky  Linguist, philosopher at MIT  1956, papers on information and grammar Types of formal grammar  Generative grammar  Analytical grammar © University of LiverpoolCOMP319slide 9

Generative formal grammars Generative grammars: A set of rules by which all possible strings in a language to be described can be generated by successively rewriting strings starting from a designated start symbol. In effect it formalises an algorithm that generates strings in the language. © University of LiverpoolCOMP319slide 10

Analytic formal grammars Analytic grammars: A set of rules that assumes an arbitrary string as input, and which successively reduces or analyses that string to yield a final boolean “yes/no” that indicates whether that string is a member of the language described by the grammar In effect a parser or recogniser for a language © University of LiverpoolCOMP319slide 11

Generative grammar components Chomsky’s definition – essentially for linguistics but perfect for formal computing grammars; consists of the following components:  A finite set N of nonterminal symbols  A finite set  of terminal symbols disjoint from N  A finite set P of production rules where a rule is of the form: string in (   N)* → string in (   N)*  A symbol S in N that is identified as the start symbol © University of LiverpoolCOMP319slide 12

Generative grammar definition A language of a formal grammar: G = (N, ,P, S) Is denoted by L(G) And is defined as all those strings over  such that can be generated by starting from the symbol S and then applying P until no more nonterminal symbols are present © University of LiverpoolCOMP319slide 13

A generative formal grammar Given the terminals {a, b}, nonterminals {S, A, B} where S is the special start symbol and Productions: S → ABS S →  (the empty string) BA → AB BS → b Bb → bb Ab → ab Aa → aa Defines all the words of the from a n b n, (i.e. n copies of a followed by n copies of b) © University of LiverpoolCOMP319slide 14

Context Free Grammars Theoretical basis of most programming languages. Easy to generate a parser using a compiler compiler. Two main approaches exist: top-down parsing e.g. LL parsers, and bottom-up parsing e.g. LR parsers. © University of LiverpoolCOMP319slide 15

LL parser Table based, top down parser for a subset of the context-free grammars (LL grammars). Parsing is Left to right, and constructs a Leftmost derivation of the sentence. LL(k) parsers use k tokens of look-ahead to parse the LL(k) grammar sentence. LL(1) grammars are popular and fast because only the next token is considered in parsing decisions. © University of LiverpoolCOMP319slide 16

Table based LL parsing © University of LiverpoolCOMP319slide 17 Input buffer: | Stack | | S Output $ | | ^ | | Parsing | | table | Architecture Consider the grammar 1.S → F 2.S → ( S + F) 3.F → 1 This has the parsing table e.g. 1 and S implies rule 1 i.e. Stack S is replaced with F and 1 is output Stack and Input same = delete Stack and Input different = error Example input ( ) $ ()1+$ S2-1-- F--3--

Table based LL parsing © University of LiverpoolCOMP319slide 18 Consider the grammar 1.S → F 2.S → ( S + F) 3.F → 1 This has the parsing table e.g. 1 and S implies rule 1 i.e. Stack S is replaced with F and 1 is output Stack and Input same = delete Stack and Input different = error Example input ( ) $ ()1+$ S2-1-- F--3-- inputstackactionoutput (S$parse ( S : 22 ((S + F)$( ( delete2 1S + F)$parse 1 S : 121 1F + F)$parse 1 F : F)$1 1 delete F)$+ + delete213 1F)$parse 1 F : )$1 1 delete2133 ))$) ) delete2133 $$stop2133

Parse Tree

Left Right Parser Bottom up parser for context-free grammars used by many program language compilers Parsing is Left to right, and produces a Rightmost derivation. LR(k) parsers uses k tokens of look-ahead. LR(1) is the most common type of parser used by many programming languages. Usually always generated using a parser generator which constructs the parsing table; e.g. Simple LR parser (SLR), Look Ahead LR (LALR) e.g. Yacc, Canonical LR. © University of LiverpoolCOMP319slide 20

Left Right parser example.. Rules... 1) E → E * B (2) E → E + B (3) E → B (4) B → 0 (5) B → 1 © University of LiverpoolCOMP319slide 21

Left Right parser example © University of LiverpoolCOMP319slide 22

Re-writing Rewriting is a general process involving strings and alphabets. Classified according to what is rewritten e.g. strings, terms, graphs, etc. A rewrite system is a set of equations that characterises a system of computation that provides one method of automating theorem proving and is based on use of rewrite rules. Examples of practical systems that use this approach includes the software Mathematica. © University of LiverpoolCOMP319slide 23

Re-writing logic example ! ! A = A// eliminate double negative !(A AND B) = !A OR !B // de-morgan © University of LiverpoolCOMP319slide 24

Re-writing in Mathematica (Wolfram) © University of LiverpoolCOMP319slide 25

L-systems Named after Aristid Lindenmeyer ( ) a Swedish theoretical biologist and botanist who worked at the University of Utrecht (Netherlands) Are a formal grammar used to model the growth and morphology of plants and animals In plant and animal modelling a special form, the parametric L-system is used – based on rewriting. Because of their recursive, parallel, and unlimited nature they lead to concepts of self- similarity and fractional dimension and fractal- like forms. © University of LiverpoolCOMP319slide 26

L-system structure The basic system is identical to formal grammars: G = {V, S, Ω, P} where G is the grammar defined V (the alphabet) a set of symbols that can be replaced by (variables) S is a set of symbols that remain fixed (constants) Ω(start, axiom or initiator) a string from V, the initial state P is a set of rules or productions defining the ways variables can be replaced by constants and other variables. Each rule, consists of a LHS (predecessor) and RHS (successor) © University of LiverpoolCOMP319slide 27

© University of LiverpoolCOMP319slide 28 Slide 28 Example 1: Fibonacci numbers V: A B C: none Ω : A P: p1: A → B p2: B → AB N=0 A N=1 → B N=2 → AB N=3 → BAB N=4 → ABBAB N=5 → BABABBAB N=6 → ABBABBABABBAB N=7 → BABABBAB... Counting lengths we get : 1,1,2,3,5,8,13,21,... The Fibonacci numbers

© University of Liverpool COMP319slide 29 Slide 29 Example 2: Algal growth V: A B C: none Ω : A P: p1: A → AB p2: B → A N=0 A → AB N=1 → ABA N=2 → ABAAB N=3 → ABAABABA

© University of LiverpoolCOMP319slide 30 COMP319 Software Engineering II Example 3: Koch snowflake V: F C: none Ω : F P: p1: F → F+F-F- F+F N=0 F N=1 → F+F-F-F+F N=2 → F+F-F-F+F+F... N=3 etc

Example 4: 3D Hilbert curve © University of LiverpoolCOMP319slide 31

Example 5: Branching © University of LiverpoolCOMP319slide 32